Ferromagnets are substances which undergo magnetostriction and also have a high magnetic permeability, a definite saturation point, significant residual magnetism and hysteresis. Examples of ferromagnets include iron, cobalt, nickel and their compounds, several rare earths and their compounds, and a class of copper-manganese-tin alloys known as "Heusler" alloys. The atomic magnetic moments in ferromagnets are arranged in volumes consisting of parallel alignments of magnetic moments called domains.
This is in contrast to diamagnets and paramagnets which do not have domains. Diamagnets such as copper, silver and gold as well as superconductors further do not have net permanent atomic magnetic moments in the absence of a magnetic field, because individual electronic spins cancel each other out. Although paramagnets such as tin, aluminum and tungsten actually have net permanent atomic magnetic moments, the moments point in random directions, leading to no net magnetization in the absence of a magnetic field.
The basic principles of magnetism as well as the mathematical relations between the various magnetic properties are further discussed on pages 1-24 in U.S. Pat. application Ser. No. 08/953,192 filed on Oct. 17, 1997 (hereinafter "Application") which are hereby incorporated by reference.
The energy of a material is known to be minimized when domains are present. Specifically, the magnetocrystalline anisotropy energy (anisotropy), which represents only a portion of the total energy in ferromagnetic materials, is minimized in the presence of domains because of the property known as magnetostriction. Magnetostriction refers to a change in the dimensions of the ferromagnetic body caused by a change in its state of magnetization. Specifically, magnetostriction is the fractional change in length along the direction of magnetization.
The origin of magnetostriction can be described following Neel's theory in terms of an exchange interaction between atomic magnetic moments. (L. Neel, Anisotropie Magnetique Superficielle et Surstructures D'orientation, J. Phys. Rad., 15, 227 (1954)), which is hereby incorporated by reference in its entirety. The exchange interaction, however, does not contribute to anisotropy since it only depends on the angle between moments and not the angle between the moments and the lattice. Anisotropy depends on the direction the moments make relative to the bond direction and results from a strong coupling between the "orbit" of an electron and the lattice of a crystal. Specifically, the electronic orbital moment interacts with the lattice by way of overlapping electron wave functions of the neighboring atoms. The "spin" of the electron is also indirectly coupled to the lattice due to a weak Russell-Saunders spin-orbit coupling. It is clear, therefore, that there is a relation between magnetization direction and the crystal axes.
In fact, the most important aspect of the relation between anisotropy energy (or anisotropy) and magnetostriction is the strain dependence of the anisotropy. Inherent in the anisotropy is a dependence on the state of strain of a crystal. Therefore, magnetostriction occurs because the anisotropy energy is minimized when the lattice deforms. The strain dependent part of the anisotropy energy is simply the sum of a magnetoelastic energy E.sub.me and an elastic energy, E.sub.el when total energy minimization or equilibrium is reached for finite strains.
Beginning with this known relationship, further mathematical relationships can be developed to describe the equilibrium strain state of a magnetostrictive material. It is also possible to determine the strain in a particular direction as well as the saturation magnetostriction in a particular direction. However, because any random domain distribution can be a "demagnetized state," it is not possible to define the demagnetized state uniquely. As a result, the magnetic saturation is used as the reference state since saturation is uniquely defined. To determine the magnetic saturation in a given direction, therefore, the saturation strain with magnetization, M, applied perpendicular to the given direction is typically subtracted from the saturation strain along that same given direction.
Therefore, the higher the anisotropy, the higher the field strength one needs to achieve magnetic saturation, and hence the higher the magnetostriction. For example, while the magnetostrictions of the 3d transition ferromagnets iron, cobalt and nickel are of the order of 20 ppm, the magnetostrictions of some of the rare earth-transition metal alloys are of the order of 1500 ppm or greater.
Material composition is one of the most important factors which influences magnetostriction. It is known, for example, that the magnetostriction of terbium, dysprosium and iron alloys is very sensitive to the relative concentrations of rare earth, and of rare earth to iron. Material microstructure also plays an important role in magnetostriction as well because the grain size and defect density are significant factors to consider for large strain.
Clearly, the mechanical state of a magnetostrictive material is intimately related to its magnetic state. This relationship can also be mathematically quantified with thermodynamic equations of state, or the magnetomechanical constitutive equations. It is important to keep in mind, however, that the magnetomechanical constitutive equations are defined for linear, reversible, nonhysteretic behavior, so they are approximations which are often impractical for ferromagnetic materials.
The development of various mathematical relationships relating to magnetostriction which focus on a "static" characterization of magnetostrictive materials is discussed in greater detail on pages 25 to 38 of Application which are hereby incorporated by reference. Of particular interest are two related parameters, i.e., an "energy conversion coefficient," k.sup.2, which represents the ratio of mechanical energy to magnetic energy, and a "magnetomechanical coupling coefficient," k, which characterizes the magnetostrictive properties of materials, both static and dynamic. The most common method of static characterization for ferromagnetic materials including magnetostrictive materials is the hysteresis loop or curve, with the traditional curve being induction versus applied field and an alternate curve showing magnetostriction versus applied field. Pages 39-43 of Application, which are hereby incorporated by reference, discuss and show plots (FIGS. 2.2 and 2.3) of these two types of hysteresis loops for a material known as Terfenol-D, which is discussed in more detail below.
Although the mathematical relationships noted above focus on "static" characterization, it is possible to derive engineering parameters to characterize the "dynamic" frequency response of a magnetostrictive material. This is important because magnetostrictive materials for transducers and actuators are usually operated under ac conditions. Some of the mathematical relationships which focus on a "dynamic" characterization of magnetostrictive materials are discussed on pages 41 and 44-49 of Application, which are hereby incorporated by reference. It is important to note that under ac conditions, k.sup.2 will be lower due to eddy current losses, such that k decreases with frequency. Another plot (FIG. 2.4) in Application shows the complex relative permeability for Terfenol-D under zero applied bias field. An offset relative to the expected theoretical component can be seen, which may be due to eddy currents or losses discussed below. These known mathematical relationships, however, do not fully characterize the properties of magnetostrictive materials, and a better characterization of these materials is necessary if they are to be fully understood and utilized in dynamic conditions.
As noted above, materials with high magnetostrictions are potentially useful in many dynamic or "ac" applications, including transducers and actuators/sensors for adaptive vibration control of sensitive optical equipment, automobiles, aircraft engines, and micropositioning of satellites. When operating under ac conditions, one of the most important considerations is the complex permeability of the material. During ac conditions, permeability is a complex quantity with a real part and an imaginary part. That is, it has an in-phase component and an out-of-phase component. The imaginary part, I, is the component of the induction, B, which is out of phase in reference to the applied magnetic field, H. The imaginary part is related to power losses in the material which partly result from the eddy currents that dissipate energy and limit the field penetration.
It is known that dynamic uses require a material having a high frequency response. The problem with many magnetostrictive materials is that they do not have the requisite high frequency response needed, primarily because they possess large eddy current losses. Such losses are apparent from a decrease in the effective permeability with frequency at fixed conductivity. Therefore, these materials have a permeability which remains constant for only a narrow frequency range.
For example, although the room temperature magnetostriction of a rare earth-iron alloy known as Terfenol-D (or Terfenol) along a cube diagonal [111], i.e., .lambda..sub.111, has a magnitude of 1640 ppm, the conductivity of Terfenol is only 60.times.10.sup.8 ohm-m, which significantly reduces the magnetic field penetration. (Terfenol refers to two pseudo-binary alloys with the compositions Tb.sub..3 Dy.sub..7 Fe.sub.2 and Tb.sub..27 Dy.sub..73 Fe.sub.2.). Therefore, as the frequency increases, the skin depth of the Terfenol decreases, e.g., at 50 kHz the skin depth of Terfenol is typically 1.5 mm. (Further details, including a plot of skin depth versus log frequency for Terfenol are discussed on pages 74-76 of Application, which are hereby incorporated by reference.) The reduced skin depth limits the diameter of a Terfenol rod that can usefully be employed at any given frequency and consequently restricts the load-bearing capability of an actuator. It also results in a drastically reduced energy conversion efficiency as the frequency of excitation increases.
It should be noted that the research that led to the development of Terfenol was undertaken because of the unusual magnetic and magnetoelastic properties of the rare earths. Further details on the rare earths as well as the development, microstructure and magnetization process of Terfenol are discussed on pages 62-73 of Application which are hereby incorporated by reference.
There is a need, therefore, to extend the useful operating range of magnetostrictive materials such as Terfenol. Presently, as noted above, performance is limited by skin depth, which depends on the conductivity of the material, and decreases with frequency. What is needed is a material which retains its high magnetostrictive properties and skin depth in dynamic applications, but also has minimal eddy current losses and a high frequency response. There is also a need to understand and fully characterize the properties of magnetostrictive materials, such as Terfenol, under ac conditions.